TEXTOS DE PATRICE LUMUMBA
EL FUTURO DE UN CONTINENTE
Homodyne detection is a very versatile technique based on interferometry that can be used to measure specific attributes of the field that would be otherwise unaccessible. It is performed by combining the field ˆa, in the role of the signal to be detected, with a stronger reference field ˆaLO, acting as a local oscillator. The name (derived from the ancient Greek words hom´os, “same”, and d´ynamis, “power”) hints that the two fields oscillate at the same frequency, !o, and to guarantee temporal coherence the same source is often used for both. The intensity of the local oscillator is typically high to enhance the interferometric component, and can thus be modelled as a classical coherent field↵LO.
The signal and the local oscillator are combined at a beam splitter (as shown in Fig. 2.7), resulting in the two output fields
ˆ
d1=t↵LO+rˆa, (2.87)
ˆ
d2=r⇤↵LO t⇤ˆa, (2.88)
where r and t are the reflection and transmission coefficients of the beam splitter, related by the conditions |r|2+|t|2 = 1 and r⇤t+rt⇤ = 0 [49]. A photodetector after the first output port of the beam splitter records an intensity proportional to
The interference of the two fields is represented by the last term on the right-hand side of the equation. This makes the reading depend not only on the intensity but also on the complex amplitude of the signal. The large amplitude of the local oscillator enhances the interference and could make even weak signals easier to detect. As the intensity of the field grows quadratically with the amplitude, however, the measurement in the case of the single photodetector of Fig. 2.7a might become overly tainted and information on the signal could be swamped by the local oscillator instead of being boosted. This problem is easily circumvented by the use of another photodetector on the second output port (as in Fig. 2.7b), whose measurement would be proportional to
hdˆ†2dˆ2i=|r|2|↵LO|2+|t|2hˆa†ˆai rt⇤h↵LO⇤ aˆ ↵LOˆa†i. (2.90)
The two readings can then be subtracted, analogically or digitally, to obtain
hdˆ†2dˆ2i hdˆ†1dˆ1i= 2|r|2 1 |↵LO|2+ 1 2|r|2 haˆ†aˆi
+ 2|r|
q
1 |r|2e i⇡2h↵⇤
LOaˆ ↵LOˆa†i. (2.91)
Here the condition r⇤t+rt⇤ = 0 was used to infer that the relative phase between r
and t is ⇡/2 [49]. The terms related to the intensity of the two fields are then easily eliminated by choosing |r|2 =|t|2 = 1/2, corresponding to a 50:50 beam splitter. The subtracted output in this case is proportional to
hdˆ†2dˆ2i hdˆ†1dˆ1i=|↵LO|he i(✓LO+⇡2)ˆa+ei(✓LO+⇡2)ˆa†i, (2.92)
where the phase of the local oscillator was specified by writing↵LO=|↵LO|ei✓LO in the
frame rotating at the optical frequency of both fields. This is precisely the quadrature ˆ
X✓LO+⇡/2 of the signal field, as defined in Eq. 2.32. Therefore, any quadrature of the field can be revealed by homodyne detection after an appropriate choice of the local oscillator’s phase, whereas the amplitude of the local oscillator acts as an e↵ective gain for the measurement.
Heterodyne detection (from the ancient Greekh´eteros, “di↵erent”) is based on very similar principles to those of homodyne detection, with the only di↵erence being in the local oscillator frequency, !LO, which is not restricted to be the same as that of the signal. The di↵erence induces a beating component, and the measurement of
𝑎 ˆLO 𝑎 ˆ 𝑑ˆ1 𝑑ˆ2 (b) 𝑎ˆLO 𝑎 ˆ 𝑑ˆ1 𝑑ˆ2 (a)
Figure 2.7: Schematic of a homodyne/heterodyne detection setup. (a)Combination and de- tection of the signal field with a reference local oscillator. (b)A clean quadrature measurement is obtained after subtracting the read-outs of photodetectors at both ports.
hdˆ†2dˆ2i hdˆ†1dˆ1iresults centred around a carrier frequency|!LO !o|:
hdˆ†2dˆ2i hdˆ†1dˆ1i=|↵LO|he i(✓LO+⇡2)e i(!LO !o)ˆa+ei(✓LO+⇡2)ei(!LO !o)aˆ†i. (2.93)
In the presence of low-frequency background noise, this feature can be very useful as the information coming from the signal may be shifted to a di↵erent spectral region, clear of contamination.
Both detection methods are very e↵ective for the measurement of the quadrature of the signal field, whether this is another coherent state like the local oscillator, or a single photon, or even a squeezed vacuum state. It should be emphasized one more time, however, that the analysis presented assumes a high-power local oscillator. Although this is sufficient for the scope of this thesis, a more complete treatment is required [27] to extend the concept to general interference of two quantum fields.